Alloying-Induced Structural Transition in the Promising Thermoelectric Compound CaAgSb

AMX Zintl compounds, crystallizing in several closely related layered structures, have recently garnered attention due to their exciting thermoelectric properties. In this study, we show that orthorhombic CaAgSb can be alloyed with hexagonal CaAgBi to achieve a solid solution with a structural transformation at x ∼ 0.8. This transition can be seen as a switch from three-dimensional (3D) to two-dimensional (2D) covalent bonding in which the interlayer M–X bond distances expand while the in-plane M–X distances contract. Measurements of the elastic moduli reveal that CaAgSb1–xBix becomes softer with increasing Bi content, with the exception of a steplike 10% stiffening observed at the 3D-to-2D phase transition. Thermoelectric transport measurements reveal promising Hall mobility and a peak zT of 0.47 at 620 K for intrinsic CaAgSb, which is higher than those in previous reports for unmodified CaAgSb. However, alloying with Bi was found to increase the hole concentration beyond the optimal value, effectively lowering the zT. Interestingly, analysis of the thermal conductivity and electrical conductivity suggests that the Bi-rich alloys are low Lorenz-number (L) materials, with estimated values of L well below the nondegenerate limit of L = 1.5 × 10–8 W Ω K–2, in spite of the metallic-like transport properties. A low Lorenz number decouples lattice and electronic thermal conductivities, providing greater flexibility for enhancing thermoelectric properties.


SI 1: Lattice parameter conversion from hexagonal to orthorhombic
To uniformly compare the changes in unit cell size as a function of composition, the lower symmetry orthorhombic lattice parameters are used.Figure S1 shows the hexagonal P63mc crystal structure, where the blue lines represent the hexagonal P63mc unit cell, while the red lines represent orthorhombic Pnma unit cell.The arrows represent respective crystallographic axes.The equations on the right can be applied to convert the hexagonal lattice parameters into the orthorhombic lattice parameters.
Figure S1: The orthorhombic unit cell (red) mapped on the hexagonal crystal structure.The hexagonal lattice parameters can be converted to orthorhombic lattice parameters using the equations shown above.Note that the volume of the orthorhombic cell, Vorth = 2 x Vhex SI 2: Rietveld refinement and lattice parameters Rietveld refinements were performed on as-SPS'ed pellets to calculate lattice parameters.The results are summarized for samples in the orthorhombic crystal structure (Pnma) and for samples with the hexagonal crystal structure (P63mc) in SI Tables 1 and 2 respectively.SI Figure 2 shows the refinement fits for each sample.The evolution of shear and bulk modulus is shown in Figure S3 as a function of composition.Shear modulus follows the same trend as Young's modulus, showing ~7% stiffening at the structural transition.Bulk Modulus for the hexagonal samples is significantly lower than that in the orthorhombic structure.For samples x = 0.0, 0.1, 0.6, and 1.0, elastic constants were determined as a function of temperature.The results show monotonous softening as a function of temperature, as seen in Figure S4.

SI 4: Debye temperature and glassy limit:
Table S3 shows the density, Rule-of-Mixture(ROM) relative densities, and volume per atom (Va) calculated from lattice parameters for SPS'ed pellets.The longitudinal and transverse speeds of sound, νL and νT respectively, are calculated using RUS.The mean speed of sound (νm) is calculated using the equation 1,2 : The Debye temperature is then calculated using the equation 2 : The Cahill minimum lattice thermal conductivity, also called the glassy or amorphous limit, is calculated using the equation 3 : The diffuson model by Agne et al. was used to calculate the minimum thermal conductivity κ diff using the equation 4 : Table S3: Select properties across the solid solution CaAgSb1-xBix.
x Density (g/cm    For heavy elements such as Sb and Bi, spin-orbit coupling (SOC) can play a role in the band structure and subsequent properties.To probe the effects, DFT calculations were carried out to include spin-orbit interactions.Figure S7 shows a comparison between PBE and PBE+SOC band structures.There is little-to-no difference between the two band structures in CaAgSb.However, in the case of CaAgBi, SOC reveals inverted band crossings in the Γ -M and Γ -K directions.The results are consistent with what was seen by Sasmal et al 5 .SI 8: Electronic and lattice thermal conductivity estimated using the SPB model: Traditionally, electronic thermal conductivity is calculated using the Weidemann-Franz law, where the Lorenz number has certain assumptions built into it.Using the SPB-model and assuming that acoustic deformation potential scattering is dominant, Lorenz number can be extrapolated using the equation   = 1.5 +  (− || 116 ⁄ ) , which has been used extensively in literature.Figure S8 shows the electronic thermal conductivities (κE) for all values of x in CaAgSb1-xBix, followed by the lattice thermal conductivity (κL), calculated by subtracting electronic contributions from total thermal conductivity.Eminently, the lattice thermal conductivity is found to be negative for Bi-rich samples through this approach. ).The results are shown based on heating data only.
SI 9: Landauer formalism to get Lorenz number: In the diffusive limit Landauer formalism, the transport distribution function, G(E), is given as:

𝜏(𝐸). 𝑔(𝐸)
where ν(E), τ(E), and g(E) denote the energy-dependent carrier velocity, relaxation time, and density of states (DOS) respectively.For this work, we assume that the carrier relaxation time τ(E) scales inversely to the magnitude of the DOS, which leads to the equation: where τ0 is the relaxation time scaling factor.This turns the G(E) equation to: At a given temperature, the integral for Seebeck coefficient is solved for experimental Seebeck coefficient to determine the Fermi energy level EF.The Seebeck integral can be written as:

SI 10: Isothermal Lorenz number:
The Weidemann-Franz law can also be written as   =   +     , which represents the equation of a straight line.The isothermal Lorenz number model assumes that at a given temperature, the proportionality constant remains constant for a given series of materials.As such, the proportionality constant and therefore the Lorenz number can be calculated from linear fit of total thermal conductivity vs electrical conductivity plot.Linear fits of the data at different temperatures are shown in Figure S10, where Lorenz number (LIsothermal) is calculated from the slope.Linear extrapolation of L was done as a function of temperature and the values were put in the Weidemann-Franz law equation (  = .. ℎ ,) to calculate electronic contributions to thermal conductivity.Subtracting κE from κTotal, κL was calculated, as shown in SI Figure 10.LIsothermal was used to calculate electronic thermal conductivity and lattice thermal conductivity using the Weidemann-Franz law.Positive values of κL were found for all samples over the entire temperature range, as shown in Figure S11.

SI 11: Electronic transport along different directions:
In this study, the thermal transport has been measured parallel to the pressing direction while resistivity is measured perpendicular to the pressing direction.To ensure that the properties are consistent in all directions, electronic transport properties were measured along both directions for CaAgBi-end member.A tall sample was prepared and cut along the two directions.Figure S13 shows that hall mobility and carrier concentration are similar along the two directions, despite some scatter.The resistivity varies between parallel and perpendicular direction by 10 to 20 %, which translates to a similar difference in electronic contribution to thermal conductivity.When LSPB was used to calculate electronic and lattice contributions to thermal conductivity, despite the difference in directional electrical conductivity, lattice thermal conductivity was found to be negative throughout the entire temperature range.The electronic and lattice contributions to thermal conductivity were calculated using LSPB and the Weidemann-Franz Law, and the 10-20% difference is seen in both results across the two directions.Despite the difference, κL remains negative across the entire temperature range for CaAgBi.Note that κTotal is calculated from thermal diffusivity measured parallel to the pressing direction.

Figure S3 :
Figure S3: Evolution of the room temperature (a) shear and (b) bulk elastic moduli as a function of composition, x.The bulk modulus decreases slightly in the hexagonal phase, which is the result of the Poisson ratio decreasing.

SI 5 :
Comparison of thermoelectric transport properties of CaAgSbPrior to this work, the thermoelectric properties of undoped CaAgSb were reported in reference 30, where the samples were synthesized through a Pb-flux.Here, we compare the thermoelectric properties of CaAgSb in this work to the flux grown CaAgSb results in reference 30.

Figure S5 :
Figure S5: TE properties of CaAgSb found in this work when compared to that reported in ref 30.(a) Seebeck coefficient and (b) resistivity are seen to increase, suggesting lower of carrier concentration in the present work.This caused a decrease in (c) thermal conductivity and an overall increase in (d) zT by almost an order of magnitude.Lattice thermal conductivity was calculated using the Wiedemann-Franz Law where Lorenz number was determined using the empirical equation based on the Seebeck coefficient, LSPB.

Figure
Figure S6: (a) Hall Mobility, (b) carrier concentration, (c) resistivity, (d) Seebeck coefficient, and (e) total thermal conductivity as a function of temperature.Solid symbols represent heating data and hollow, patterned symbols represent cooling data.

Figure S7 :
Figure S7: Band structures for CaAgSb show no difference (a) without SOC or (b) with SOC.For CaAgBi, there is a large difference between (c) without SOC and (d) with SOC band structures, as band inversion is seen when SOC is included in the calculations.

Figure S9 :
Figure S9: Seebeck coefficient vs. energy integrand used to find the position of EF at 400 K for (a) CaAgSb and (b) CaAgBi using experimental values of Seebeck coefficient.(c) Density of states and (d) carrier velocity calculated from DFT shows the position of the determined EF at 400 K.For CaAgSb, EF lies at 0.16 eV below the valence band maximum, while that for CaAgBi lies at 0.46 eV.

Figure S10 :
Figure S10: (a) τ0(T) for the two end members and (b) Isothermal relaxation time as a function of energy at 400 K. (c) Electronic thermal conductivity calculated for both end members are shown along with experimental total thermal conductivity as a function of temperature.

Figure S11 :
Figure S11: Experimental total thermal conductivity vs. electrical conductivity, used to determine values of the Isothermal Lorenz number at (a) 323 K, (b) 423 K, (c) 523 K, and (d) 623 K.According to the Wiedemann-Franz Law, the graphs are fit to the equation   =   +      and the Lorenz number, LIsothermal is calculated from the slope at each temperature.

Figure
Figure S12: (a) Electronic contributions and (b) lattice contributions to thermal conductivity calculated using the isothermal Lorenz number model.Results were taken based on heating data only.

Figure S13 :
Figure S13: To assess anisotropy effects, van der Pauw measurements were conducted in both parallel and perpendicular directions to the pressing axes for CaAgBi.Both (a) hall mobility and (b) carrier concentrations were found to be almost identical within the scatter.(c) Resistivity results show deviations of 10 to 20% between the two directions.(d)The electronic and lattice contributions to thermal conductivity were calculated using LSPB and the Weidemann-Franz Law, and the 10-20% difference is seen in both results across the two directions.Despite the difference, κL remains negative across the entire temperature range for CaAgBi.Note that κTotal is calculated from thermal diffusivity measured parallel to the pressing direction.

Table S1 :
Rietveld refinement results for samples in the orthorhombic (Pnma) crystal structure:

Table S2 :
Phase quantity and lattice parameters calculated through Rietveld refinement for samples with multiphase and hexagonal (P63mc) crystal structure: